Complex Number Calculator

Picture a standard coordinate grid. Every point on that grid has an x-coordinate (horizontal) and a y-coordinate (vertical). A complex number a + bi is simply a label for the point at position (a, b) on this grid — called the complex plane. The real part a moves you left or right, and the imaginary part b moves you up or down. Nothing more exotic than that. The unusual part is what happens when you multiply two of these points together.

Adding and subtracting complex numbers works exactly like vector addition: combine the real parts, combine the imaginary parts, done. Multiplication is where the structure of complex numbers reveals itself. The rule i² = -1 means that squaring the imaginary unit flips you from the vertical axis to the negative real axis — a 90-degree rotation. Multiplying two complex numbers in general rotates one by the angle of the other and scales it by the other's distance from the origin. This rotation-and-scaling interpretation is why complex numbers show up in every field where cycles, oscillations, or rotations appear.

Division by a complex number uses a technique called multiplying by the conjugate. The conjugate of c + di is c - di. Multiplying the denominator by its own conjugate eliminates the imaginary part from the denominator: (c + di)(c - di) = c² + d², a plain real number. The numerator gets the same conjugate treatment, and the result splits cleanly into a real and imaginary component. This is the only legal path to dividing complex numbers in rectangular form without going through polar form first.

Use this tool any time you need to combine two complex numbers in a single arithmetic step — checking homework, verifying a step in an AC circuit analysis, confirming a transfer function computation, or working through a signal processing problem. It is appropriate for any level from precalculus through graduate engineering as long as the numbers are in rectangular form.

This tool is not appropriate when you need to raise a complex number to a power, find roots of complex numbers, or compute functions like the complex exponential or complex logarithm. For repeated multiplication, polar form is more efficient — multiply magnitudes and add angles instead of applying FOIL multiple times. For roots, De Moivre's theorem gives a direct formula that this rectangular calculator does not implement.

The tool also does not handle symbolic inputs. Every entry must be a decimal number. If your problem involves variables or exact fractions, work through the algebra by hand and enter the final decimal values, or use a computer algebra system for exact symbolic answers.

Forgetting that i² = -1, not +1. The most common algebra error in complex multiplication is treating i² as positive one, which flips the sign of the real part of the result. If your multiplication result has a real part that seems too large or has the wrong sign, check whether you correctly subtracted bd from ac rather than adding it.

Entering the imaginary coefficient without the i. The input fields ask for the coefficient of i — the number that multiplies i. For 5 + 3i, enter 3 in the imaginary field, not 3i or the full expression. Entering a text string like '3i' into a number field will be read as not-a-number and the tool will reject it.

Expecting division to always produce a clean result. Division of two complex numbers with integer components almost never produces integer components in the output. The denominator c² + d² is shared by both the real and imaginary parts of the result, and only cancels cleanly in special cases. If your division result shows decimals, that is correct — do not round prematurely before using the result in further calculations.

Every operation follows from two rules: the distributive property of algebra, and the identity i² = -1. Addition is direct: real parts add to real parts, imaginary parts add to imaginary parts. (a + bi) + (c + di) = (a + c) + (b + d)i. Subtraction is the same structure with minus signs: (a - c) + (b - d)i.

Multiplication expands like any binomial product. (a + bi)(c + di) = ac + adi + bci + bdi². Substituting i² = -1 turns the last term into -bd, giving real part (ac - bd) and imaginary part (ad + bc). For the example 2 + 3i multiplied by 1 + 4i: real = (2)(1) - (3)(4), imaginary = (2)(4) + (3)(1), yielding 4 + 6i when those inputs are used.

Division introduces the conjugate. To compute (a + bi)/(c + di), multiply top and bottom by (c - di). The denominator becomes 7.211103 squared — no, more precisely, c² + d², a real denominator. The numerator becomes (ac + bd) + (bc - ad)i. Divide each component by the same denominator to get the final rectangular result. Magnitude uses the Pythagorean theorem: |z| = √(a² + b²), treating the complex number as a point in the plane. Argument uses θ = atan2(b, a) to return the correct angle across all four quadrants, measured in degrees.

The rectangular form formulas assume exact arithmetic, but floating-point computation introduces rounding at roughly 15- 16 significant digits. When subtracting two nearly equal complex numbers, catastrophic cancellation can make the result far less accurate than the inputs suggest — a known limitation of any floating-point implementation. For high-precision work, verify that your inputs differ sufficiently that cancellation is not an issue.

Division in rectangular form is also numerically less stable than the equivalent polar calculation (divide magnitudes, subtract angles) when the denominator has very small magnitude. If c² + d² is close to the machine epsilon, the result components may be wildly inflated. In production code, this case is typically handled by scaling the denominator before dividing.